@article{Frame:1978, key = "Frame 1978", author = "J. S. Frame", title = "A short proof of quadratic reciprocity", journal = AMM, volume = 85, year = 1978, pages = "818-819"} @article{Shallit:1990, key = "Shallit 1990", author = "J. O. Shallit", title = "On the worst case of three algorithms for computing the {Jacobi} symbol", journal = JSC, year = 1990, volume = 10, pages = "593-610"} @book{Hua:1982, key = "Hua 1982", author = "L. K. Hua", title = "Introduction to Number Theory", publisher = SV, address = NY, year = 1982} @article{Estes&Pall:1973, key = "Estes and Pall 1973", author = "D. R. Estes and G. Pall", title = "A reconsideration of {Legendre-Jacobi} symbols", journal = JNT, volume = 5, year = 1973, pages = "433-434"} @article{Cartier:1970, key = "Cartier 1970", author = "P. Cartier", title = "Sur une {g\'en\'eralisation} des symboles de {Legendre-Jacobi}", journal = EM, volume = 16, year = 1970, pages = "31-48"} @article{Graham&Yao&Yao:1978, key = "R. Graham, Yao, and Yao 1978", author = "R. L. Graham and A. C.-C. Yao and F.-F. Yao", title = "Addition chains with multiplicative cost", journal = DM, volume = 23, year = 1978, pages = "115-119"} @article{Kannan&Lipton:1986, key = "Kannan and Lipton 1986", author = "R. Kannan and R. J. Lipton", title = "Polynomial-time algorithm for the orbit problem", journal = JACM, volume = 33, year = 1986, pages = "808-821"} @article{Collins&Reddy&Sloane:1978, key = "S. Collins, Reddy, and Sloane 1978", author = "S. Collins and S. M. Reddy and N. J. A. Sloane", title = "Elementary Problem 2704", journal = AMM, volume = 85, year = 1978, pages = "198"} @article{Ecker:1979, key = "Ecker 1979", author = "M. W. Ecker", title = "Elementary Problem 2773", journal = AMM, volume = 86, year = 1979, pages = "393"} @article{Euler:1761, key = "Euler 1761", author = "L. Euler", title = "Theoremata circa residua ex divisione potestatum relicta", journal = NCASP, volume = "7 {\rm (1758/9)}", year = "1761", pages = "49-82", note = "Reprinted in {\it Opera Omnia}, Ser.~1, Vol.~2, pp.~493--518"} @article{Blakley&Borosh:1983, key = "Blakley and Borosh 1983", author = "G. R. Blakley and I. Borosh", title = "Modular arithmetic of iterated powers", journal = CMA, volume = 9, year = 1983, pages = "567-581"} @article{Dawson:1994, key = "Dawson 1994", author = "R. J. M. Dawson", title = "Towers of powers modulo $m$", journal = CMJ, volume = 25, year = 1994, pages = "22-28"} @article{Radke:1970, key = "Radke 1970", author = "C. E. Radke", title = "The use of quadratic residue research", journal = CACM, volume = 13, year = 1970, pages = "103-105"} @article{Maurer:1968, key = "W. Maurer 1968", author = "W. D. Maurer", title = "An improved hash code for scatter storage", journal = CACM, volume = 11, year = 1968, pages = "35-38"} @inproceedings{Egecioglu&Koc:1990, key = "{{E\~{g}ecio\~{g}lu} and Ko{\c c} 1990}", author = "O. {\noopsort{Egecioglu}}{E\~{g}ecio\~{g}lu} and C. K. Ko{\c c}", title = "Fast modular exponentiation", booktitle = "Proc. 1990 Bilkent Int'l. Conf. New Trends in Communication, Control, and Signal Processing", publisher = "Elsevier", year = 1990, volume = 1, pages = "188-194"} @article{Yao:1976, key = "Yao 1976", author = "A. C. Yao", title = "On the evaluation of powers", journal = SIAMJC, volume = 5, year = 1976, pages = "100-103"} @article{Brauer:1939, key = "Brauer 1939", author = "A. Brauer", title = "On addition chains", journal = BAMS, volume = 45, year = 1939, pages = "736-739"} @article{Downey&Leong&Sethi:1981, key = "Downey, Leong, and Sethi 1981", author = "P. Downey and B. Leong and R. Sethi", title = "Computing sequences with addition chains", journal = SIAMJC, volume = 10, year = 1981, pages = "638-646"} @article{Erdos:1960, key = "{Erd{\H{o}}s} 1960", author = "P. {Erd{\H{o}}s}", title = "Remarks on number theory {III}: on addition chains", journal = AA, volume = 6, year = 1960, pages = "77-81"} @article{Dobkin&Lipton:1980, key = "Dobkin and Lipton 1980", author = "D. Dobkin and R. J. Lipton", title = "Addition chain methods for the evaluation of specific polynomials", journal = SIAMJC, volume = 9, year = 1980, pages = "121-125"} @article{Lipton&Dobkin:1976, key = "Lipton and Dobkin 1976", author = "R. J. Lipton and D. Dobkin", title = "Complexity measures and hierarchies for the evaluation of integers and polynomials", journal = TCS, volume = 3, year = 1976, pages = "349-357"} @inproceedings{Pippenger:1976, key = "Pippenger 1976", author = "N. Pippenger", title = "On the evaluation of powers and related problems", booktitle = FOCS76, year = 1976, pages = "258-263"} @article{Semba:1983, key = "Semba 1983", author = "I. Semba", title = "Systematic method for determining the number of multiplications required to compute $x^m$, where $m$ is a positive integer", journal = JIP, volume = 6, year = 1983, pages = "31-33"} @article{Scholz:1937, key = "Scholz 1937", author = "A. Scholz", title = "Aufgabe 253", journal = JDMV, volume = 47, year = 1937, pages = "Supplement, pp. 41-42"} @article{Allouche&Shallit:1992, key = "Allouche and Shallit 1992", author = "J.-P. Allouche and J. O. Shallit", title = "The ring of $k$--regular sequences", journal = TCS, volume = 98, year = 1992, pages = "163-187"} @article{Gioia&Subbarao&Sugunama:1962, key = "Gioia, Subbarao, and Sugunama 1962", author = "A. A. Gioia and M. V. Subbarao and M. Sugunama", title = "The {Scholz-Brauer} problem in addition chains", journal = DMJ, volume = 29, year = 1962, pages = "481-487"} @article{Thurber:1976, key = "Thurber 1976", author = "E. G. Thurber", title = "Addition chains and solutions of $\ell(2n) = \ell(n)$ and $\ell(2^n-1) = n + \ell(n) - 1$", journal = DM, volume = 16, year = 1976, pages = "279-289"} @article{Thurber:1973a, key = "Thurber 1973a", author = "E. G. Thurber", title = "The {Scholz-Brauer} problem on addition chains", journal = PJM, volume = 49, year = "{\noopsort{1973a}}1973", pages = "229-242"} @article{Thurber:1973b, key = "Thurber 1973b", author = "E. G. Thurber", title = "On addition chains $\ell(mn) \leq \ell(n) - b$ and lower bounds for $c(r)$", journal = DMJ, volume = 40, year = "{\noopsort{1973b}}1973", pages = "907-913"} @article{Cottrell:1973, key = "Cottrell 1973", author = "A. Cottrell", title = "A lower bound for the {Scholz-Brauer} problem", journal = NAMS, volume = 20, year = 1973, pages = "A-476", note = "Abstract 73T-A200"} @article{Straus:1964, key = "Straus 1964", author = "E. G. Straus", title = "Solution to Problem 5125", journal = AMM, volume = 71, year = 1964, pages = "807-808"} @article{Vegh:1975, key = "Vegh 1975", author = "E. Vegh", title = "A note on addition chains", journal = JCTA, volume = 19, year = 1975, pages = "117-118"} @article{Tsai&Chin:1987, key = "Tsai and Chin 1987", author = "Y. H. Tsai and Y. H. Chin", title = "A study of some addition chain problems", journal = IJCM, volume = 22, year = 1987, pages = "117-134"} @article{Olivos:1981, key = "Olivos 1981", author = "J. Olivos", title = "On vectorial addition chains", journal = JA, volume = 2, year = 1981, pages = "13-21"} @article{Leeuwen:1977, key = "van Leeuwen 1977", author = "Leeuwen, J. van", title = "An extension of {Hansen's} theorem for star chains", journal = JFRAM, volume = 295, year = 1977, pages = "202-207"} @inproceedings{Gioia&Subbarao:1978, key = "Gioia and Subbarao 1978", author = "A. A. Gioia and M. V. Subbarao", title = "The {Scholz-Brauer} problem in addition chains {II}", booktitle = "Proc. Eight Manitoba Conference on Numerical Math. and Computing", editor = "D. McCarthy and H. C. Williams", year = 1978, pages = "251-274", note = "(= {\it Congr. Numer.} {\bf XXII})"} @incollection{Bos&Coster:1990, key = "Bos and Coster 1990", author = "J. Bos and M. Coster", title = "Addition chain heuristics", booktitle = CRYPTO89, editor = "G. Brassard", publisher = SV, year = 1990, series = LNICS, volume = 435, pages = "400-407"} @techreport{Tompa:1991, key = "Tompa 1991", author = "M. Tompa", title = "Lecture notes on probabilistic algorithms and pseudorandom generators", number = "91-07-05", institution = "Department of Computer Science and Engineering, University of Washington", year = 1991, month = "July"} @article{Karp&Rabin:1987, key = "Karp and Rabin 1987", author = "R. M. Karp and M. O. Rabin", title = "Efficient randomized pattern-matching algorithms", journal = IBMJ, volume = 31, year = 1987, pages = "249-260"} @article{Zolotarev:1872, key = "Zolotarev 1872", author = "G. Zolotarev", title = "Nouvelle {d\'emonstration} de la loi de {r\'eciprocit\'e} de {Legendre}", journal = NAM, volume = 11, year = 1872, pages = "354-362"} @book{Pieper:1978, key = "Pieper 1978", author = "H. Pieper", title = "Variationen {\"uber} ein zahlentheoretisches {Thema} von {Carl} {Friedrich} {Gauss}", publisher = "{Birkh\"auser}", address = "Basel", year = 1978} @techreport{Brlek&Mallette:1992, key = "Brlek and Mallette 1992", author = "S. Brlek and R. Mallette", title = "Sur le calcul des chaines d'additions optimales", institution = "{D\'epartement} de {math\'ematiques} et d'informatique, {Universit\'e} du {Qu\'ebec} {\`a} {Montr\'eal}", number = 173, month = "February", year = 1992, note = "Abbreviated version in J. Labelle and J.-G. Penaud, eds., {\it Atelier de Combinatoire Franco-Qu\'ebecois, 6-7 mai 1991}, Publications du LaCIM {\bf 10} (1992), 71--85"} @article{Prodinger&Tichy:1983, key = "Prodinger and Tichy 1983", author = "H. Prodinger and R. F. Tichy", title = "{\"Uber} ein zahlentheoretisches {Problem} aus der {Informatik}", journal = OAW, volume = 192, year = 1983, pages = "385-396"} @incollection{Brickell&Gordon&McCurley&Wilson:1993, key = "Brickell, Gordon, McCurley, and Wilson 1993", author = "E. F. Brickell and D. M. Gordon and K. S. McCurley and D. B. Wilson", title = "Fast exponentiation with precomputation", booktitle = EUROCRYPT92, editor = "R. A. Rueppel", series = LNICS, volume = 658, year = 1993, publisher = SV, pages = "200-207"} @incollection{Iwamura&Matsumoto&Imai:1993a, key = "Iwamura, Matsumoto, and Imai 1993a", author = "K. Iwamura and T. Matsumoto and H. Imai", title = "Systolic-arrays for modular exponentiation using {Montgomery} method", booktitle = EUROCRYPT92, editor = "R. A. Rueppel", series = LNICS, volume = 658, year = "{\noopsort{1993a}}1993", publisher = SV, pages = "477-481"} @inproceedings{Kawamura&Hirano:1988, key = "Kawamura and Hirano 1988", author = "S.-I. Kawamura and K. Hirano", title = "A fast modular arithmetic algorithm using a residue table", booktitle = EUROCRYPT88, volume = 330, series = LNICS, publisher = SV, editor = "C. G. {G\"unther}", year = 1988, pages = "245-250"} @phdthesis{CooperA:1926, key = "A. Cooper 1926", author = "A. E. Cooper", title = "A Topical History of the Theory of Quadratic Residues", month = "June", year = 1926, school = "University of Chicago"} @article{Aiello&Subbarao:1993, key = "Aiello and Subbarao 1993", author = "W. Aiello and M. V. Subbarao", title = "A conjecture in addition chains related to {Scholz}'s conjecture", journal = MC, volume = 61, year = 1993, pages = "17-23; S1-S6"} @article{LenstraHW:1993, key = "H. W. Lenstra 1993", author = "Lenstra, Jr., H. W.", title = "Generating units modulo an odd integer by addition and subtraction", journal = AA, volume = 64, year = 1993, pages = "383-388"} @article{Lagrange:1771, key = "Lagrange 1771", author = "J. L. Lagrange", title = "{D\'emonstration} d'un {th\'eor\`eme} nouveau concernant les nombres premiers", journal = "Nouveaux {M\'emoires} de l'{Acad\'emie} Royale des Sciences et Belles-Lettres de Berlin", year = 1771, pages = "125-137", note = "Reprinted in {\it Oeuvres}, Vol.~3, 1869, pp.~425-438"} @article{Egecioglu&Koc:1994, key = "{{E\~{g}ecio\~{g}lu} and Ko{\c c} 1990}", author = "O. {\noopsort{Egecioglu}}{E\~{g}ecio\~{g}lu} and C. K. Ko{\c c}", title = "Exponentiation using canonical recoding", journal = TCS, year = 1994, volume = 129, pages = "407-417"} @unpublished{deMelo&Svaiter:1995, key = "de Melo and Svaiter 1995", author = "Melo, W. de and Svaiter, B. F.", title = "The cost of computing integers", note = "To appear, {\it Proc.\ Amer.\ Math.\ Soc.}", year = 1995} @article{Thurber:1993, key = "Thurber 1993", author = "E. G. Thurber", title = "Addition chains -- an erratic sequence", journal = DM, volume = 122, year = 1993, pages = "287-305"} @article{Utz:1953, key = "Utz 1953", author = "W. R. Utz", title = "A note on the {Scholz-Brauer} problem in addition chains", journal = PAMS, volume = 4, year = 1953, pages = "462-463"}